Basic Math Examples

${(\frac{7 r - 7 y}{r + y})}^{2} \div (49 r^{2} - 49 y^{2})$

Step 1

Rewrite the division as a fraction.

$\frac{{(\frac{7 r - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2

Factor $7$ out of $7 r - 7 y$ .

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Step 2.1

Factor $7$ out of $7 r$ .

$\frac{{(\frac{7 (r) - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2.2

Factor $7$ out of $- 7 y$ .

$\frac{{(\frac{7 (r) + 7 (- y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2.3

Factor $7$ out of $7 (r) + 7 (- y)$ .

$\frac{{(\frac{7 (r - y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to $\frac{7 (r - y)}{r + y}$ .

$\frac{\frac{{(7 (r - y))}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the product rule to $7 (r - y)$ .

$\frac{\frac{7^{2} {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 3.2.2

Raise $7$ to the power of $2$ .

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $49$ out of $49 r^{2} - 49 y^{2}$ .

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Step 4.1.1

Factor $49$ out of $49 r^{2}$ .

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) - 49 y^{2}}$

Step 4.1.2

Factor $49$ out of $- 49 y^{2}$ .

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) + 49 (- y^{2})}$

Step 4.1.3

Factor $49$ out of $49 (r^{2}) + 49 (- y^{2})$ .

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2} - y^{2})}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = y$ .

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r + y) (r - y)}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}} \cdot \frac{1}{49 (r + y) (r - y)}$

Step 6

Combine.

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{2} (49 (r + y) (r - y))}$

Step 7

Multiply ${(r + y)}^{2}$ by $r + y$ by adding the exponents.

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Step 7.1

Move $r + y$ .

$\frac{49 {(r - y)}^{2} \cdot 1}{(r + y) {(r + y)}^{2} (49 (r - y))}$

Step 7.2

Multiply $r + y$ by ${(r + y)}^{2}$ .

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Step 7.2.1

Raise $r + y$ to the power of $1$ .

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1} {(r + y)}^{2} (49 (r - y))}$

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1 + 2} (49 (r - y))}$

Step 7.3

Add $1$ and $2$ .

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{3} (49 (r - y))}$

Step 8

Multiply $49 {(r - y)}^{2}$ by $1$ .

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{3} (49 (r - y))}$

Step 9

Cancel the common factor of $49$ .

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Step 9.1

Cancel the common factor.

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{3} (49 (r - y))}$

Step 9.2

Rewrite the expression.

$\frac{{(r - y)}^{2}}{{(r + y)}^{3} (r - y)}$

Step 10

Cancel the common factor of ${(r - y)}^{2}$ and $r - y$ .

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Step 10.1

Factor $r - y$ out of ${(r - y)}^{2}$ .

$\frac{(r - y) (r - y)}{{(r + y)}^{3} (r - y)}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $r - y$ out of ${(r + y)}^{3} (r - y)$ .

$\frac{(r - y) (r - y)}{(r - y) {(r + y)}^{3}}$

Step 10.2.2

Cancel the common factor.

$\frac{(r - y) (r - y)}{(r - y) {(r + y)}^{3}}$

Step 10.2.3

Rewrite the expression.

$\frac{r - y}{{(r + y)}^{3}}$